Geometry & Topology

Eigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrations

Fei Yu

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Abstract

Inspired by the Katz–Mazur theorem on crystalline cohomology and by the numerical experiments of Eskin, Kontsevich and Zorich, we conjecture that the polygon of the Lyapunov spectrum lies above (or on) the Harder–Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using the works of Atiyah and Bott, Forni, and Möller. We obtain several applications to Teichmüller dynamics conditional on the conjecture.

Article information

Source
Geom. Topol., Volume 22, Number 4 (2018), 2253-2298.

Dates
Received: 12 October 2016
Accepted: 21 August 2017
First available in Project Euclid: 13 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1523584822

Digital Object Identifier
doi:10.2140/gt.2018.22.2253

Mathematical Reviews number (MathSciNet)
MR3784521

Zentralblatt MATH identifier
06864337

Subjects
Primary: 14H10: Families, moduli (algebraic) 30F60: Teichmüller theory [See also 32G15] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]

Keywords
moduli space of Riemann surface Teichmüller geodesic flow eigenvalue of curvature Lyapunov exponent Harder–Narasimhan filtration

Citation

Yu, Fei. Eigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrations. Geom. Topol. 22 (2018), no. 4, 2253--2298. doi:10.2140/gt.2018.22.2253. https://projecteuclid.org/euclid.gt/1523584822


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